Expand the following:
f(n) = (n+0) (n-1) (n-2) (n-3) (n-4)
(n-3) (n-4) = (n) (n) + (n) (-4) + (-3) (n) + (-3) (-4) = n^2-4 n-3 n+12 = n^2-7 n+12:
f(n) = n n^2-7 n+12 (n-1) (n-2)
(n-1) (n-2) = (n) (n) + (n) (-2) + (-1) (n) + (-1) (-2) = n^2-2 n-n+2 = n^2-3 n+2:
f(n) = n n^2-3 n+2 (n^2-7 n+12)
| | | | n^2 | - | 3 n | + | 2
| | | | n^2 | - | 7 n | + | 12
| | | | 12 n^2 | - | 36 n | + | 24
| | -7 n^3 | + | 21 n^2 | - | 14 n | + | 0
n^4 | - | 3 n^3 | + | 2 n^2 | + | 0 n | + | 0
n^4 | - | 10 n^3 | + | 35 n^2 | - | 50 n | + | 24:
f(n) = n n^4-10 n^3+35 n^2-50 n+24
n (n^4-10 n^3+35 n^2-50 n+24) = n n^4+n (-10 n^3)+n (35 n^2)+n (-50 n)+n 24:
f(n) = n n^4-10 n n^3+35 n n^2-50 n n+24 n
n (-50) n = -50 n^2:
f(n) = n n^4-10 n n^3+35 n n^2+-50 n^2+24 n
n×35 n^2 = n^(1+2)×35:
f(n) = n n^4-10 n n^3+35 n^(1+2)-50 n^2+24 n
1+2 = 3:
f(n) = n n^4-10 n n^3+35 n^3-50 n^2+24 n
n (-10) n^3 = n^(1+3) (-10):
f(n) = n n^4+-10 n^(1+3)+35 n^3-50 n^2+24 n
1+3 = 4:
f(n) = n n^4-10 n^4+35 n^3-50 n^2+24 n
n n^4 = n^(1+4):
f(n) = n^(1+4)-10 n^4+35 n^3-50 n^2+24 n
1+4 = 5:
Answer: |f(n) = n^5 - 10n^4 + 35n^3 - 50n^2 + 24n
